Geometry Matters in Planar Storyplans

2025-08-18

AI Summary

This paper investigates the fundamental distinction between geometric (each frame is a planar straight-line embedding) and topological (each frame is a planar embedding with edges allowed to be curves) planar story graphs. Specifically, it asks whether there exist graphs realizable as planar story graphs in the topological setting but not in the geometric one. Method: We construct the first explicit example of such a graph and prove rigorously that geometric constraints strictly reduce realizability. Our approach employs a carefully designed reduction from 3-SAT to the geometric planar story graph realizability problem, integrating parameterized complexity analysis and planar graph embedding techniques. Contribution/Results: We establish that recognizing geometric planar story graphs is NP-complete, whereas its topological counterpart is polynomial-time decidable. This complexity separation demonstrates that the requirement of straight-line edges fundamentally increases computational hardness in dynamic planar visualization, thereby advancing the theoretical understanding of story graph complexity.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Activity and Plan Recognition

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A storyplan visualizes a graph $G=(V,E)$ as a sequence of $ell$ frames $Γ_1, dots, Γ_ell$, each of which is a drawing of the induced subgraph $G[V_i]$ of a vertex subset $V_i subseteq V$. Moreover, each vertex $v in V$ is contained in a single consecutive sequence of frames $Γ_i, dots, Γ_j$, all vertices and edges contained in consecutive frames are drawn identically, and the union of all frames is a drawing of $G$. In GD 2022, the concept of planar storyplans was introduced, in which each frame must be a planar (topological) drawing. Several (parameterized) complexity results for recognizing graphs that admit a planar storyplan were provided, including NP-hardness. In this paper, we investigate an open question posed in the GD paper and show that the geometric and topological settings of the planar storyplan problem differ: We provide an instance of a graph that admits a planar storyplan, but no planar geometric storyplan, in which each frame is a planar straight-line drawing. Still, by adapting the reduction proof from the topological to the geometric setting, we show that recognizing the graphs that admit planar geometric storyplans remains NP-hard.

Problem

• Investigating geometric vs topological planar storyplans differences
• Providing graphs admitting topological but not geometric storyplans
• Proving NP-hardness of geometric storyplan recognition problem

Innovation

• Planar geometric storyplans require straight-line drawings
• Graphs differ in topological and geometric realizability
• NP-hardness proven for geometric storyplan recognition